I think the first time this hit me I was looking at some software which allowed you to generate the Mandelbrot set and zoom into any level of detail you wanted, and thinking "how can this be possible? Does it really go on forever?" But it wasn't just seeing the level of detail go on infinitely that drove the significance home, but rather when I finally looked up the algorithm that generates the Mandelbrot set, and saw how simple it was. That was what made me first think, "Yeah, we probably can't know everything there is to know."